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Abstract

Background

The objective of this study is to develop a comprehensive model of the electromechanical
behavior of the rat ventricular myocyte to investigate the various factors influencing
its contractile response.

Methods

Here, we couple a model of Ca2 + dynamics described in our previous work, with a well-known model of contractile mechanics
developed by Rice, Wang, Bers and de Tombe to develop a composite multiphysics model
of excitation-contraction coupling. This comprehensive cell model is studied under
voltage clamp (VC) conditions, since it allows to focus our study on the elaborate
Ca2 + signaling system that controls the contractile mechanism.

Results

We examine the role of various factors influencing cellular contractile response.
In particular, direct factors such as the amount of activator Ca2 + available to trigger contraction and the type of mechanical load applied (resulting
in isosarcometric, isometric or unloaded contraction) are investigated. We also study
the impact of temperature (22 to 38°C) on myofilament contractile response. The critical
role of myofilament Ca2 + sensitivity in modulating developed force is likewise studied, as is the indirect
coupling of intracellular contractile mechanism with the plasma membrane via the Na + /Ca2 + exchanger (NCX). Finally, we demonstrate a key linear relationship between the rate
of contraction and relaxation, which is shown here to be intrinsically coupled over
the full range of physiological perturbations.

Conclusions

Extensive testing of the composite model elucidates the importance of various direct
and indirect modulatory influences on cellular twitch response with wide agreement
with measured data on all accounts. Thus, the model provides mechanistic insights
into whole-cell responses to a wide variety of testing approaches used in studies
of cardiac myofilament contractility that have appeared in the literature over the
past several decades.

Ca2 + is an extremely important and highly versatile second messenger in cardiac cells,
which plays a crucial role not only in excitation-contraction (E-C) coupling but also
in excitation-transcription coupling [2]. Various inter-connected Ca2 + signalling pathways help preserve the integrity of the cellular Ca2 + system despite any disturbances (e.g., changes in stimulation frequency or inotropic
state). A key role for the dyadic Ca2 + release system is the formation of an adequate myoplasmic Ca2 + transient that can serve as an input driving signal for the intracellular contractile
machinery (the myofilaments). The resultant contractile response is conditioned by
a number of additional factors that include the mechanical load; sarcomere equilibrium
length; myofilament Ca2 + sensitivity; and the temperature. Although it is well-known that the contractile response
of a cell is a function of the stimulation frequency (its force-frequency response
(FFR)), this study is limited to an investigation at 5 Hz (unless otherwise specified),
a physiologically relevant heart rate for a rat ventricular myocyte.

Computational methods

All simulations and analysis were performed on a 2.8GHz Intel® CoreTM2 Duo CPU-based computer using Microsoft Windows XP operating system. The sarcolemmal
membrane charge balance equations, the Ca2 + material balance equations in the myoplasm and SR, and the force balance equations
describing the model for myofilament contraction constitute a set of 93 ordinary differential
equations (ODEs). A fixed-step Merson-modified Runge-Kutta 4th-order numerical integration
scheme [3] was used to solve this set of 1st-order differential equations (ODE) describing the
dynamic model. The free Ca2 + concentration in the dyad is governed by the time courses of the Ca2 + fluxes through Ca2 + transport systems, as well as by the time course of Ca2 + binding to Ca2 + buffers present in the junction [4]. Description of the spatio-temporal dynamics of calcium transients in the dyad triggered
by Ca2 + stimulus (basis of CICR) requires calculation of the partial differential equations
(PDE) of the whole reaction-diffusion system. Formation and dissipation of Ca2 + gradients around an open channel (DHP-sensitive and Ry-sensitive channels in the dyad)
is assumed instantaneous as was validated for microsecond timescale and nanoscopic
space by Naraghi and Neher [5]. Local Ca2 + concentration in the vicinity of open channels (located on opposing boundaries of
the dyadic space) was calculated as the steady state gradient around a point source
[6]. The Ca2 + concentration increments from individual channels at each point in space were assumed
to be additive [5,7]. The software kernel follows the changes in the state of trigger and release channels
together with variables like membrane voltage and spatial Ca2 + concentration to calculate the instantaneous rate constants and estimate the duration
of transient events. Crank [8] discusses diffusion problems in a two-phase heterogeneous medium and shows that diffusion
through a system of barriers (RyR feet structures in the dyadic cleft space) can be
approximated by diffusion in the same region without barriers but with a reduced effective
diffusion coefficient. We hence take this approach in modeling the Ca2 + diffusion by solving the 2-D Laplacian equation (Krishna et al. [4], Appendix A3, Eq. 140) in the DCU without explicitly accounting for local potential
fields. More specifically, an explicit finite difference scheme was used to solve
these Laplacian equations describing Ca2 + -diffusion in the dyadic space analogous to the method detailed in Smith et al. [9]. Specifically, a radial symmetry is employed in solving the PDE in the dyadic volume
allowing the solution to be computed in a rectangular cross-section discretized into
a 20 by 20 cartesian grid. The spatial step size used in the r and z-direction (Figure
1B, Krishna et al. [4]) was 10 nm and 0.76 nm respectively (Table two, Krishna et al. [4]). We use the method of lines (discretization in space) to solve the PDE. The full
set of ODEs and finite difference equations are solved simultaneously to obtain the
complete solution. Execution of a single cycle which translates to 200 ms at 5 Hz
took 21 seconds with a time step of 1μs. Results were visualized using Matlab (Mathworks, Natick, MA) and Origin (OriginLab
Corp., Northampton, MA).

Model development

Our objective was to develop an integrated model of the rat ventricular cell under
voltage clamp conditions, which includes the description of various Ca2 + signalling pathways in the dyadic space, the myoplasmic medium and the sarcoplasmic
reticulum (Figure 1, adopted from Krishna et al. [4]) as well as a comprehensive coupled mechanical system describing the contractile
machinery responsible for force generation.

Electrochemical description of Ca2 +sub-system

Our model for the electrochemical description of the cell consists of an electrical-equivalent
model for the cell membrane and a fluid-compartment model describing the flux of ionic
species between the extracellular and several intracellular compartments (cell cytosol,
SR and the dyadic coupling unit (DCU), in which resides the mechanistic basis of CICR).
The DCU is described as a controller-actuator mechanism, internally stabilized by
negative feedback control of the unit’s two diametrically-opposed Ca2 + channels (trigger-channel and release-channel). It releases Ca2 + flux into the cytoplasm and is in turn enclosed within a negative feedback loop involving
the SERCA pump, regulating Ca2 + myo. A detailed description of the membrane classification, channel and exchanger distribution
as well as the various fluid compartments involved is given in Krishna et al. [4]. Our model for the electrochemical description of the Ca2 + sub-system is based on our previous work [4] with the following modifications: (a) the rate constants used to model L-type Ca2 + current (ICa,L), maximum Na + /Ca2 + exchanger current (INaCa), maximum plasma membrane Ca2 + -ATPase current (IPMCA) and the maximal uptake rate of the SERCA pump are Q10 adjusted using the values given in Table 1 to model temperature dependence; (b) The effect of cAMP-mediated β-adrenergic stimulation is modeled by allowing the relative regulatory activity of
PKA to be a function of the available isoproterenol concentration (Appendix, Equations
57).

Mechanical description of myofilament contractile system

Our model for cardiac contractile mechanics (Figure 2) is based on the approximate model of cooperative activation and crossbridge cycling
reported by Rice et al. [13] with the following modifications: (a) the first-order rate constants for the transformation
of the troponin/tropomyosin regulatory complex (outside the single overlap region
between the thick and thin filaments) from a crossbridge non-permitting state to a
crossbridge permitting state and vice-versa are chosen as 500 s-1 and 50 s-1 respectively
in order to reproduce results reported by Rice et al. [13]; (b) the β-adrenergic agonist isoproterenol (ISO) is known to cause a decrease in myofilament
Ca2 + sensitivity as a result of protein kinase A (PKA) mediated phosphorylation of troponin
I [15,16] at Ser23/Ser24. Specifically, a two-state Markovian model is added to allow isoproterenol-dependent
PKA-mediated interaction between troponin I (TnI) and the Ca2 + -binding regulatory site on troponin. As shown in Figure 2, state TnIu, which denotes the unphosphorylated form of TnI, modulates the Ca2 + affinity of the regulatory site on troponin. This is in agreement with the findings
of Messer et al. [17]. We model the effects of ISO by allowing the cumulative activation rate constant
for Ca2 + -binding to the troponin regulatory site to be a function of unphosphorylated TnI
(TnIu), the availability of which is in turn dependent on the amount of ISO present (Appendix,
Equations 15); (c) the large Q10 values used by Rice et al. (Qfapp, Qhf, Qhb and Qgxb, Table 1, [13]) are decreased from 6.25 to 2.25 in order to reproduce temperature dependence of
peak force developed in intact thin rat ventricular trabeculae [14]. Although a calmodulin (CaM) mediated pathway has been reported [18] to be responsible for modulation of myofibrillar Ca2 + -sensitivity (implying a possible CaM mediated role for Ca-dependent kinases or phosphatases
in regulating myofilament contractility, particularly in frequency dependent acceleration
of relaxation), we refrain from modeling this effect as the molecular mechanisms involved
remain unresolved.

Figure 2.Model for cardiac electro-mechanics. Cooperative activation and crossbridge cycling. Model for cardiac electro-mechanics
and force generation based on Rice et al. [13] shows states in non-permissive and permissive confirmations of the regulatory proteins.
The permissive state transitions into a pre-rotated (XB_PreR) state having a strongly
bound crossbridge with the head extended. The transition to the post-rotated (XB_PostR)
force-generating state represents the isomerization to induce strain in the extensible
neck region. Activation is triggered by the fraction of Ca2 + bound troponin regulatory units (T_Reg) which sets the rate constant for transition
between the non-permissive (N_XB) to permissive (P_XB) confirmation of the regulatory
protein using a strong nonlinearity function to indicate cooperativity. The model
assumes that troponin for regulation has affinity set by the thin filament overlap.
The affinity for apparent Ca2 + binding (used to perturb Ca2 + myo), not only depends on thin filament overlap but also increases as crossbridges strongly
bind to populate the pre-rotated and post-rotated states. The regulatory and apparent
Ca2 + binding terms are calculated separately to avoid a global feedback from strongly-bound
crossbridges to Ca2 + binding causing nonphysiological Ca2 + sensitivity [13].

Testing protocol

We employ our coupled multiphysics model describing the electrochemical as well as
the mechanical subsystems to study cellular contraction emphasizing the various modulatory
influences that are at play. We begin with a detailed analysis of different types
of twitch response to better understand the influence of various factors such as sarcomere
length, peak Ca2 + myo and the stiffness of the contractile element in cell shortening, followed by a comparative
study of these twitch responses. The negative feedback of cellular contraction on
the myoplasmic Ca2 + -transient [19,20] is also investigated. We then perform an idealized virtual experiment similar to
that carried out in an experimental study [21] to uncover the regulation of cell contraction by Na + /Ca2 + exchange, and in the process identifying the role of the SERCA pump in facilitating
this effect. We then model the effects of temperature [22] on cardiac contractile response. This is followed by a study identifying the role
of myofilament Ca2 + sensitivity as a key factor influencing the degree of cell shortening. In particular,
the effect of β-adrenergic agonist isoproterenol, which is known to cause a decrease in myofilament
Ca2 + sensitivity [15], is investigated. Hence, we have developed an integrated multiphysics model of rat
ventricular cell electromechanics and now seek to study its response to various tests
prescribed by elements of the virtual protocol above. In doing so, we hope to identify
and clarify the role played by key factors involved in modulation of the cell’s contractile
response.

Results

From our modeling standpoint, the dyadic coupling unit (DCU) as defined by Krishna
et al. [4] is a fundamental element involved in the mechanism of CICR. This previous study described
the control features of this unit, as well as its interaction with the SERCA pump
and free sarcolemmal pumps and exchangers to achieve a homeostatic regulation of myoplasmic
Ca2 + concentration. We now extend our voltage clamp studies to address the subject of force
generation following CICR, starting with the classical twitch responses below.

Twitch responses

A twitch response, which is a brief contractile response of a cardiac cell elicited
by dynamically changing activator Ca2 + , is a commonly used experimental characterization. Following are the three types
of recordings commonly used to quantify force generation in isolated cardiac cells:
(1) isosarcometric contraction which is experimentally obtained by incorporating feedback
sarcomere length (SL) control using laser diffraction techniques [23-25]; (2) unloaded contraction where the cell is allowed to contract freely; and (3) isometric
contraction where overall muscle length is fixed, but sarcomere length is not controlled,
allowing considerable internal shortening as a result of compliant end connections
(series elastic element). All the twitch studies are carried out at 22.5°C [13] to be consistent with the data cited. Under all three loading conditions, the cell
is subjected to conditioning train of steady state (100 cycles) voltage clamp pulses
at 5 Hz followed by a 0.8 s rest interval (allowing decay to a zero force resting
state) which is subsequently followed by a single test twitch for which a transient
lasting 1 s is obtained. The voltage clamp protocol used is a 50 ms step pulse to
10 mv from a holding potential of -40 mv. In this study, we report a normalized force
with a maximum value of 1 possible under optimal conditions such as high Ca2 + myo, isosarcometric loading (SL = 2.3 μm) and low temperature allowing maximum overlap of thick and thin filaments.

Isosarcometric contraction

As mentioned above, in isosarcometric contraction, SL is maintained constant via external
feedback control. We simulate this type of contraction under two distinct conditions:
modulation of developed force by fixed changes in (A) SL, and (B) myoplasmic Ca2 + concentration. In case ‘A’ while SL is fixed at different values, the input Ca2 + -transient is kept identical (elicited by the standard 50 ms voltage clamp pulse at
5 Hz). In case ‘B’ while the level of activator Ca2 + is modulated, SL is kept fixed at 2.3 μm.

(A) The steady state force-Ca2 + (F-Ca) relationship shown in Figure 3A-i exhibits a leftward shift and an increase in developed maximum plateau force as
SL is clamped at increasing lengths. This leftward shift results from an increase
in myofilament Ca2 + sensitivity as SL is increased. Figure 3A-ii shows the temporal course of normalized force as SL is changed in steps from
1.8 to 2.3 μm. The waveshape of standard Ca2 + -transient is overlaid in dotted lines in this figure. Although an increase in SL
(traces marked + to ∗) does not cause a large variation in the time to peak force
(TTP), it does result in an increase in peak force magnitude and twitch duration as
the result of an increase in myofilament Ca2 + sensitivity. These characteristics show a strong correspondence with measured data
from rat ventricular myocytes tested at similar (∼ 22.5°C) temperatures [14,24,26]. The correlation coefficient of the speed of contraction and relaxation has been
experimentally observed [27] to be very close (> 0.98). The inset in Figure 3A-ii is a plot of the rate of relaxation (reciprocal of time taken for 50% sarcomere
relaxation (RT50)) versus rate of contraction (reciprocal of time taken for peak sarcomere contraction
(TTP)) for increasing SL. This linear relationship highlights contraction-relaxation
coupling, and represents a key intrinsic property of the contractile myofilaments
[27]. Figure 3A-iii shows the phase plots of self normalized force versus the instantaneous Ca2 + concentration in the cytosol for increasing SL (traces marked + to ∗) overlayed with
two steady state F-Ca relationships corresponding to SL = 1.8 μm (+) and SL = 2.3 μm (∗). The assessment of dynamic and steady-state Ca2 + relationships allows better analysis of the phase-plane loops of force versus Ca2 + . The active twitch curve is related to the steady-state values to determine, at what
isochrone the dynamic force-Ca2 + value equals that obtained in the steady-state relationship. This point of intersection
of the steady state F-Ca trace and the corresponding phase plot gives the contraction-relaxation
coupling point (CRCP, marked as ∘) from initiation of stimulation [16]. Time is implicit on the phase trajectory and at time instants prior to reaching
the critical coupling point for a particular trajectory, Ca2 + myo exceeds the value of Ca2 + predicted by the steady state F-Ca relationship. This excess favors continued sarcomere
contraction. At later time points beyond the CRCP, the developed force is greater
than that predicted by the steady state curve, which favors myofilament relaxation.

(B) Increasing background Ca2 + myo causes a leftward shift in steady state F-SL relationship as shown in Figure 3B-i. The increase in maximal plateau force with increase in background Ca2 + myo is observed to be less prominent at higher levels of activator Ca2 + in the myoplasm. In Figure 3B-ii the activator Ca2 + is varied by modulating the peak of the Ca2 + myo transient by adjusting the voltage clamp pulse duration (an increase in pulse duration
from 5 ms to 50 ms increased peak Ca2 + myo from 0.5 to 1.1 μM respectively). This protocol allows for the peak of the transient to be changed
without a significant change in the duration of the transient (Krishna et al. [4] ; Figure 4). The traces correspond to increasing peak values from 0.5 (∙) to 1.1 (∗) μM. Although similar to case with increasing SL, increasing activator Ca2 + results in a relatively non-linear increase in peak force generated. As shown in
Figure 3A-i, the steady state F-Ca relationship is characterized by a Hill function as experimentally
observed [25]. The time-to-peak force (TTP) remains relatively unaffected by the amount of activator
Ca2 + causing the twitch response. Inset in Figure 3B-ii shows the dependence of TD50 (time taken from 50% activation to 50% relaxation) on peak Ca2 + myo indicating an increase in twitch duration with increasing levels of activator Ca2 + . Figure 3B-iii shows the phase plots of self normalized force versus the instantaneous Ca2 + concentration in the myoplasm for increasing peak Ca2 + myo (traces marked ∙ to ∗) overlayed with a steady state F-Ca relationships corresponding
to SL = 2.3 μm (∗). The contraction-relaxation coupling point (∘) traverses along the F-Ca relationship
to increasing values of Ca2 + and force with increasing peak Ca2 + myo.

Unloaded contraction

The protocol for the unloaded case is as follows. The cell is not stretched with pre-load
so that the series elastic element is unattached and is therefore not in play. In
the model of Rice et al. ([13]; Figure 1), the contractile element is shunted by elastic and viscous damping elements. In
that figure, the nonlinear elastic element is characterized by a cubic force vs SL
characteristic centered about an equilibrium point (SL0=1.9 μm; F=0). In the unloaded case without stimulation, any stored energy in the system
is dissipated and SL decays to the equilibrium point on the passive force vs SL characteristic.
With electrical activation and subsequent Ca2 + release, active force is developed and SL shortening occurs against the aforementioned
passive restoring forces. Providing the same sequence of voltage clamp pulses as in
Figure 5, an identical sequence of Ca2 + -transients is produced to drive the active contractile mechanism. Figure 5A is a plot of total developed force (active and passive) as a function of peak Ca2 + myo. This net instantaneous force can become negative when the magnitude of the passive
forces exceeds that of the active component (Figure 5A). Thus, an increase in activator Ca2 + causes an increase in peak force generated, which translates into enhanced shortening.
Corresponding changes in sarcomeric length as shown in Figure 5B indicate that increasing levels of activator Ca2 + result in a decrease in time to peak (TTP declined from 156.0 ms (∙) to 70.5 ms (∗))
and an increase in the rate of relaxation (RT50 computed from time of peak decreased from 183.0 ms (∙) to 157.5 ms (∗)).

Isometric contraction

A third type of twitch can be simulated where the cell is kept at a fixed total length,
allowing it to contract in response to Ca2 + release by internal shortening of the sarcomere made possible by a non-contractile
series elastic element whose stiffness (KSE value) dictates the end compliance and
hence the degree of internal shortening. Increasing KSE values causes an increase
in maximal plateau force in steady state F-Ca relationship as shown in Figure 6A. Figure 6B shows traces for total force (sum of both passive and active force) during an isometric
twitch corresponding to KSE values increased from 1.0 to 50.0 normalized force-μm−1. With an increase in end compliance (decrease in KSE), the degree of internal shortening
increases and the total force measured at the cell end decreases, showing a delayed
peak and an increase in rate of relaxation (Figure 6B). The delayed peak occurs because the peak force is measured when the series elastic
element is at its maximum length, which occurs with greater delay with increasing
end compliance. Increasing end compliance decreases twitch duration (Figure 6B) because, as observed experimentally [28] re-lengthening hastens relaxation as a result of an increase in mean distortion of
the strongly bound crossbridge states (xXB_PreR, xXB_PostR in Figure 2) which causes a decrease in the forward rotation rate of the crossbridges (Eqn. 22,
Rice et al. [13]) and hence a faster force decline. Figure 6C shows the phase plots of self normalized force versus the instantaneous Ca2 + concentration in the myoplasm for increasing KSE values (traces marked ∙ to ∗) overlayed
with two steady state F-Ca relationships corresponding to KSE = 1.0 (∙) and KSE =
50.0 (∗). The contraction-relaxation coupling point (∘) moves to increasing values
of Ca2 + and force with increasing KSE values with the relative change in Ca2 + being smaller than force. Figure 6D shows the corresponding traces for sarcomere length during the isometric twitch.
As the KSE value is increased from 1 to 50 the decreasing compliance results in a
decline in cell shortening accompanied by a decrease in time to peak shortening from
122 ms to 92 ms (Figure 6D).

Figure 6.Comparison of twitch responses. Twitch responses - A comparison between three types of steady state twitch responses
from an isolated rat ventricular myocyte viz. isosarcometric, unloaded and isometric
(KSE = 2). Fixed SL in the isosarcometric case and the initial pre-contraction SL
in the isometric case are chosen as 2.2 μm. Equilibrium SL in the unloaded case is 1.9 μm. (A) Traces for normalized force in each of the three cases with an overlay of normalized
[Ca2 + ]myo transient responsible for the twitch and (B) Phase plots of normalized force versus the instantaneous Ca2 + concentration in the cytosol in each of the three cases. Note the overlap of ♦ and
■ in panel iii. Model generated data corresponds to an idealized rat ventricular myocyte
at 22.5°C.

Figure 7 shows the three types of simulated twitch responses studied, compared in their force
vs. time plots, as well as in their normalized force vs. Ca2 + myo phase diagrams. This plotting format aids in drawing a comparison that highlights
the unique characteristics of each loading condition. The protocol used here is a
steady state 5 Hz stimulation without a rest interval before the test twitch (unlike
Figures 3, 5 and 6). Figure 7A shows that the isosarcometric case results in maximum force development, whereas
the unloaded case records the minimum force for identical sarcomere length and initial
conditions. Figure 7B shows the phase plots of normalized force versus the instantaneous Ca2 + concentration in the cytosol constructed from model-generated data captured at steady
state (the last in a train of stimuli comprising 100 cycles at 5 Hz stimulation) from
a twitch caused by a Ca2 + myo transient resulting from a voltage clamp pulse (amplitude -40 mv to 10 mv and a duration
of 50 ms). The initial pre-contraction sarcomere length in the isometric case and
the sarcomere length clamp in the isosarcometric case are both set to 2.2 μm whereas the equilibrium length in the unloaded case is chosen as 1.9 μm [13].

Figure 7.Feedback length effects. Feedback of internal shortening on myoplasmic [Ca2 + ]myo transient. The protocol used comprises of 9 beats of isometric contraction followed
by a 0.8 s rest interval and beat 10 (shown above) when the cell is allowed to internally
shorten (∙) or held at a fixed sarcomere length of 2.2 μm (∗). (A) The sarcomere length in both the cases shows the degree of contraction when the
cell is allowed to internally shorten via a series elastic element (KSE = 2). (B) Isosarcometric case shows enhanced force when compared to the isometric case. The
inset shows corresponding steady state F-Ca relationships (C) As seen in experimental studies, the isosarcometric case shows a modest decrease
in [Ca2 + ]myo transient. Model generated data corresponds to an idealized rat ventricular myocyte
at 22.5°C.

In the case of an isosarcometric contraction the net force comprises only the active
component due to the tension generated by the sarcomere trying to contract. Phase
‘a’ in Figure 7B-i is indicative of the delay in the contractile response when compared to the [Ca2 + ]myo transient. However, soon after the [Ca2 + ]myo reaches its peak (∙), the sarcomere begins to contract, resulting in a gradual increase
in force, achieving a maximum (■) as seen in phase ‘b’ of Figure 7B-i. As active force increases, a decline in level of activator Ca2 + in the cytosol ultimately causes a recovery to the minimum contractile state (▸),
as shown in phase ‘c’ of Figure 7B-i.

Total force generated by an unloaded cell during contraction against its internal
restoring force is a combination of the active component attributed to tension generating
action of cycling crossbridges and the passive component attributed to titin and other
cytoskeletal elements. Passive force generated has a negative contribution to the
net force for SL values lower than the equilibrium length. Hence, the competition
between the active and passive components of force gives the trace in Figure 7B-ii its characteristics. During phase ‘a’ in Figure 7B-ii, as the Ca2 + level in the cytosol increases towards its peak value (∙), the sarcomere attains maximum
contraction velocity (▾), which is followed by a steep increase in net force as shown
in phase ‘b’ of Figure 7B-ii. However, decreasing sarcomere length increases the negative contribution from
the passive component of force resulting in the first transient decline shown in phase
‘c’ of Figure 7B-ii. The increased Ca2 + binding to troponin in response to the rise in activator Ca2 + in the cytosol enhances the active component of force, allowing the cell to reach
peak net contractile force (■). The fast decline in sarcomere length which causes
a rapid increase in the passive component of force results in the second transient
decline as shown in phase ‘d’ of Figure 7B-ii. However, a subsequent decrease in cell shortening soon allows an increase in
net force lasting for a short duration (while the SL reaches its minimum (♦)). The
declining Ca2 + level in the cytosol, an outcome of SR uptake, then causes a gradual recovery to the
resting state (▸).

As shown in Figure 7A, for the same activator Ca2 + , although the total muscle force generated during isometric contraction exhibits
a triphasic response similar to an isosarcometric contraction, its own unique characteristics
are a delayed time to peak (an increase from 84.5 ms to 119 ms) and relatively smaller
magnitude. As shown in phase ‘a’ of Figure 7B-iii, the increase in Ca2 + concentration is not reflected in a fast mechanical response. After [Ca2 + ]myo reaches its peak (∙), the tension in the sarcomere begins to build up (phase ‘b’
of Figure 7B-iii) although at a slower rate (compare isometric and isosarcometric traces in Figure
7A), causing a delayed time to peak (84 ms and 120 ms in isosarcometric and isometric
cases, respectively) due to the presence of the series elastic element which facilitates
slow internal shortening. The sarcomere achieves peak contraction (♦) when the total
muscle force reaches its maximum (■), following which the cell recovers back to the
minimum contractile state (▸) as shown in phase ‘c’, Figure 7B-iii. During isometric contraction (KSE = 2) the afterload (due to the series elastic
element) tracks the active component of force generated due to the tension developed
in the sarcomere while the passive component of force (attributed to titin and other
cytoskeletal elements) is small in magnitude owing to a much smaller degree of sarcomere
contraction achieved when compared to the unloaded case (compare trace marked ∗ in
Figure 5B and the trace for KSE=2 in Figure 5D).

Effect of contraction on the [Ca2 +]myo transient

The myoplasmic Ca2 + myo transient which follows SR release acts as the trigger for myofilament contraction.
However, the contracting myofilament also has a feedback effect on the shape of the
Ca2 + myo transient as a result of Ca2 + binding to the low affinity regulatory sites on troponin in the myofilament. The Ca2 + affinity of this site depends on both the sarcomere length as well as the fraction
of strongly-bound crossbridges (Eq. 37, Rice et al. [13]). The protocol used to test these effects is similar to that employed by Janssen
and de Tombe [29], wherein the cell is stimulated at 5 Hz for 9 beats under conditions of isometric
contraction (KSE=2) followed by a 0.8 s rest interval allowing decay to a zero force
resting state. This is followed by the 10th beat for which a transient lasting 1 s
is obtained. On the 10th beat, the cell is allowed to sarcometrically shorten as usual
(∙) or a sarcometric length clamp is imposed at a fixed sarcomere length of 2.2 μm (∗). At the onset of Ca2 + -activation of troponin on the 10th beat, initial conditions for the two different
loading tests (isometric and isosarcometric) are identical. Sarcomere length changes
under both loading conditions are shown in Figure 8A. As observed earlier in Figures 7B (i) and (iii) as well as in Figure 8B, isosarcometric conditions generate a larger force than isometric, due to enhanced
myofilament Ca2 + sensitivity reflected by a decrease in EC50 from 0.59 μM (∙) to 0.5 μM (∗) (steady state F-Ca relationships shown in the inset in Figure 8B) and increased Ca2 + -binding to troponin. This binding causes a small (<1%) decline in the magnitude of
the Ca2 + myo transient (Figure 8C). Subsequently, as the Ca2 + -transient starts to decay, Ca2 + dissociates from troponin, slightly decreasing the rate of decline of the Ca2 + -transient. When the falling phases of the Ca2 + -transients for the two loading conditions are compared, the isosarcometric Ca2 + -transient decays more slowly and its curve crosses over the isometric Ca2 + -transient. Although Janssen et al. [29] reported an increase in Ca2 + transient due to uncontrolled shortening, the crossover was not observed as the noise
level was too large. However, the crossover effect has been reported in other studies
on rat ventricular trabeculae [19,20] using long and short sarcomere length twitches producing larger changes in developed
force and hence the shape of the Ca2 + myo transient (>5% change in peak value).

Figure 8.Regulation of contraction via NCX. Regulation of isometric cell contraction by Na + /Ca2 + exchange. The protocol used for the isometric contraction is stimulation at 5 Hz comprising
of 30 beats when the cell undergoes unperturbed contraction (phase A) followed by
30 beats (phase B) when the Na + /Ca2 + exchanger function is partially inhibited after which the inhibition is relieved allowing
the cell to recover to a steady state control during the following 30 beats (phase
C). (A) The Na + /Ca2 + exchanger current (INaCa) showed a rapid transition between the control and inhibited phase and vice versa.
(B) Relative role of the Na + /Ca2 + exchanger and the SERCA pump. (C) Myoplasmic Ca2 + concentration. (D) Sarcomere length in response to INaCainhibition. Model generated data corresponds to an idealized rat ventricular myocyte
at 22.5°C.

Regulation of isometric cell shortening by Na +/Ca2 +exchange

During stable, steady-state operation, Ca2 + entry into the cytosol via ICa,L and SR release must exactly balance Ca2 + efflux via the sarcolemmal Na + /Ca2 + exchanger (NCX), plasma membrane Ca2 + -ATPase pump, and Ca2 + -uptake to the SR by the SERCA pump. To study the role of the NCX in isometric SL
shortening we develop a virtual experimental protocol loosely patterned after a study
on rat ventricular myocytes [21]. In our experiment, isometric contraction is stimulated by voltage clamp pulses at
5 Hz. In phase A of the experiment, 30 clamp pulses are applied resulting in regular
isometric contractions. This is immediately followed by phase B where the NCX is assumed
partially inhibited (achieved in the model by a 25% decrease in maximum Na + /Ca2 + exchange current) by rapid superfusion of a bathing solution containing a NCX inhibitor
(e.g. Phe-Arg-Cys-Arg-Ser-Phe-CONH2 (FRCRSFa) or exchanger inhibitory peptide (XIP)
which are known [30] to cause selective NCX inhibition). This is followed by phase C where rapid superfusion
with normal bathing solution completely removes the inhibitor, thus restoring exchanger
activity to initial levels. It is well known that a decrease in NCX activity (Phase
B, Figure 9A) results in an increase in SR Ca2 + content as a result of the excess Ca2 + in the cytosol being resequestered into the SR via the SERCA pump. Figure 9B elucidates the relative contribution of the Na + /Ca2 + exchanger and the SERCA pump in Ca2 + extrusion from the cytosol, showing an increase in the relative role of SERCA pump
as a result of inhibition in NCX activity. As observed experimentally [21] this enhancement in SR Ca2 + content results in an increased availability of activator Ca2 + following release (Figure 9C) which in turn enables improved cell shortening as seen by an enhancement in peak
to peak amplitude of contraction in Figure 9D (Δ increases from 0.0848 μm to 0.0977 μm) and is accompanied by a decrease in mean sarcomere length (Δm = 0.0217 μm). A similar study (not shown here) involving a sudden increase in NCX activity instead
of a decrease resulted in an opposite (a decrease in SR Ca2 + content causing diminished release and hence a decrease in degree of contraction)
symmetric response of similar magnitude indicating strong homeostatic control. Myoplasmic
Na + concentration (17 mM) remains relatively constant throughout and hence is not involved
in this indirect control of myofilament contractility. Our virtual experiment only
approximates the actual experiment in that we assume the ability to make very rapid
changes in the composition of the bathing medium relative to the time constants involved
in the response. However, the model generated responses in steady state can be compared
with measured steady state response in the presence/absence of NCX inhibition using
selective blocking agents that allow complete post-washout recovery [30].

Effect of temperature on contractile performance

Temperature is known to have a strong effect on the L-type Ca2 + current (ICa,L), the Ca2 + -transient and the contractile mechanics. One very significant effect of temperature
on whole-cell ICa,L is the pronounced increase in its rate of decline with an increase in temperature
[31,32]. Thus, with an increase from room to body temperature, peak inward trigger current
increases but the waveform becomes much narrower. Figure 10A shows model-generated ICa,Lwaveforms at temperatures between 22°C (+) and 38°C (∗) in steps of 4°C, where one
can observe the increase in peak current but also the increased rate of decline in
the trigger current waveform with an increase in temperature. Specifically, peak ICa,L at 22°C was 8.31 pA/pF compared with 15.05 pA/pF at 38°C, whereas time taken for
50% 50% ICa,L inactivation (RT50,I) decreased from 10.75 ms at 22°C to 2.95 ms 38°C (inset in Figure 10A). These indices are in general agreement with measured voltage clamp data [32,33] obtained from rat ventricular trabeculae.

Although skinned rat ventricular preparations have been used extensively in studies
of the cardiac contractile process, data from such preparations violates the assumptions
of our whole cell electromechanical model. The integrity of the plasma membrane components
and intricate Ca2 + -regulatory system are compromised to some degree, regardless of the skinning technique
used. Therefore, we have chosen to consider only data from rat ventricular myocytes
or ultra-thin rat ventricular trabeculae to help validate the model.

The decrease in integrated ICa,Land faster SR uptake with increasing temperature together cause a decline in peak
and duration of the Ca2 + -transient as shown in Figure 10B. This agrees with the experimental findings by Janssen et al. ([14]; Figure 1) on thin rat ventricular trabeculae. RT50,C corresponding to the Ca2 + -transient (time taken for 50% decline in Ca2 + myo) decreased from 22.8 ms at 22°C to 18.5 ms at 38°C (inset in Figure 10B). Myofilament Ca2 + sensitivity increases with an increase in temperature as a result of a temperature
dependent enhancement in crossbridge cycling rate [34-36]. Traces for the steady state F-Ca relationship in Figure 10C show a temperature dependent increase in myofilament Ca2 + sensitivity with no significant change in maximum plateau force. An increase in temperature
from 22°C (+) to 38°C (∗) results in a decrease in EC50 from 0.55 μM to 0.44 μM (inset in Figure 10C).

As experimentally observed [14], a decrease in peak and duration of the Ca2 + myo-transient with increasing temperature results in a corresponding overall decrease
in peak developed contractile force and twitch duration. Our simulations show changes
in ICa,L and Ca2 + myo in Figures 9A and B, and corresponding changes in developed force in Figure 10D with increasing temperature. As the temperature is increased from 22°C to 30°C,
traces for normalized force in Figure 10D show a small increase in peak amplitude despite a decline in the amount of activator
Ca2 + responsible for contraction due to a temperature dependent increase in myofilament
Ca2 + sensitivity (Figure 10C). This increase in peak developed force at low, non-physiological temperatures is
confirmed by Janssen et al. ([14]; Figure 1). However, as the temperature is increased further from 30°C to 38°C, further decreases
in peak Ca2 + myo cause strong decreases in peak developed force. Figure 10D shows that an increase in temperature was accompanied by a decrease in both time
to peak (TTP) as well as RT50,F (time taken for 50% decline in force). The inset in Figure 10D shows the linear relationship [27,37] between rate of relaxation versus rate of contraction for increasing temperature
(reciprocal of RT50,Fversus reciprocal of TTP).

Figure 10E shows phase plots of normalized force versus instantaneous Ca2 + concentration in the myoplasm for increasing temperatures (22°C (+) to 38°C (∗) in
steps of 4°C) overlayed with two steady state F-Ca relationships corresponding to
22°C (+) and 38°C (∗). The contraction-relaxation coupling point (∘) moves to decreasing
values of [Ca2 + ] and force with increasing temperature. Phase-plane analysis of normalized force
versus instantaneous Ca2 + concentration in the myoplasm reveals that, as the temperature is increased from 22°C
to 38°C, the relaxation phase moves to the right towards the corresponding steady-state
F-Ca relationship. This suggests that, with increase in temperature there is a departure
from cross-bridge kinetics being the rate-limiting step in cardiac relaxation to a
more Ca2 + -driven mechanism. As evident from Figure 10D, the inset in Figure 10E shows the temperature dependence of peak force developed indicating a moderate increase
at low (< 30°C) temperatures with a strong decline at temperatures above 30°C. Traces
for SL shortening in Figure 10F corresponding to [Ca2 + ]myo transients in panel B, indicate an overall decrease in sarcomere shortening with
increasing temperature which agrees with the trend in developed force in panel D.
Comparison of insets in Figure 10E and F shows the correlation between peak force developed and the corresponding delta
change in SL. The effect of change in temperature on myofilament contractility is
a two-stage response with feedback. Firstly, the [Ca2 + ]myo transient is temperature sensitive owing largely to the temperature dependence of
the trigger current ICa,L. Secondly, the cellular contractile machinery is highly temperature sensitive due
the strong temperature dependence of rate kinetics involved in the formation of crossbridges.
In addition, the process of crossbridge formation is known to have a small feedback
effect on the [Ca2 + ]myo transient as seen in Figure 8. As the temperature is increased from 22°C to 30°C, the temperature sensitivity of
crossbridge kinetics predominates the temperature dependence of [Ca2 + ]myo transient in determining the contractile response. However, for temperatures from
30°C to 38°C the opposite holds true.

Role of myofilament Ca2 +sensitivity

Among various factors influencing the degree of cell shortening achieved in response
to myoplasmic Ca2 + myo transient, the Ca2 + affinity of troponin C regulatory site is known to be particularly important. Here,
we studied the dependence of isometric force response on myofilament Ca2 + sensitivity (MCS). An increase in Ca2 + affinity is modeled by a 30% increase in konT (traces marked ▴ in Figure 11), the rate constant for Ca2 + -binding to troponin regulatory site. Similarly, a decrease in MCS is modeled by a
corresponding decrease in konT (traces marked ▾ in Figure 11). The steady state F-Ca relationships in Figure 11A show the gradual rightward shift in response to decreasing MCS as seen experimentally
in right ventricular trabeculae from rat in the presence of bupivacaine [38]. A positive/negative change in MCS results in the average Ca2 + myo required for half-maximal activation to decrease/increase from 0.61 μM in the control case to 0.47 μM and 0.89 μM respectively. However, the Hill coefficient was constant (4.0) showing no change
in response to modulation in MCS.

Figure 11.Effect of isoproterenol and increase in [Ca2 +]o. Role of isoproterenol and [Ca2 + ]o. (A) Time record of [Ca2 + ]myo transient in the control case as well as with 3 mM [Ca2 + ]o and 1 μM ISO. (B) Traces for normalized force recorded at steady state with an overlay of [Ca2 + ]myo transient. (C) Phase plots of normalized force versus the instantaneous Ca2 + concentration in the cytosol with an overlay of traces showing normalized steady state
force versus [Ca2 + ]myo relationship. Model generated data corresponds to an idealized rat ventricular myocyte
at 22.5°C.

Figure 11B shows traces of normalized force captured at steady state in response to a [Ca2 + ]myo transient (overlaid) in the control case as well as with modified myofilament Ca2 + sensitivity. A 30% decrease (from control value) in MCS causes a much larger change
in peak isometric force generated than a 30% increase, highlighting the nonlinear
response. Figure 11B shows that an increase in MCS causes a faster onset of the upstroke in force response
and a delay in the start of recovery as a result of enhancement in Ca2 + binding to troponin. This delay in recovery manifests in an increase in time to peak
from 119.5 ms (∙) in the control case to 124 ms (▴). Similarly, a decrease in MCS
produces an opposite effect resulting in a decrease in time to peak (112 ms (▾)).
The two distinct slopes during the upstroke in force response in Figure 11B are a result of an initial contribution of the strongly bound pre-rotated state
(XB_PreR in Figure 2) followed by the effect of increase in strongly bound post-rotated state (XB_PostR
in Figure 2). As shown in the inset in Figure 11B, an increase in MCS causes an increase in TD50 (time taken from 50% activation to 50% relaxation).

Phase plots of normalized force versus the instantaneous Ca2 + concentration in the cytosol are shown in Figure 11C. As observed experimentally [39], a decrease in myofilament Ca2 + sensitivity causes the gradient (units of Normalized force/μM) of the trajectory during the relaxation phase of the twitch contraction (marked
by thick arrows in Figure 11C) to decrease from 12.4 in the control case to 7.0. A similar but opposite effect
was observed in the rate of relaxation with a corresponding increase in MCS. A delayed
onset of the upstroke in force response as a result of a decrease in MCS (Figure 11B) causes a distinct loop at high Ca2 + concentrations (trace marked ▾ in Figure 11C). The traces for sarcomere length in Figure 11D reflect the changes in force developed, showing an increased degree of shortening
with an enhancement in MCS. Peak shortening velocity also increases with an increase
in myofilament Ca2 + sensitivity as shown in Figure 11E.

Effect of Isoproterenol: The β-adrenergic agonist isoproterenol (ISO) is known to cause a decrease in myofilament
Ca2 + sensitivity as a result of PKA-mediated phosphorylation of troponin I at Ser23/Ser24
[15,17]. However, the increase in amplitude of myoplasmic Ca2 + transient more than compensates for the decrease in Ca2 + sensitivity in order to facilitate the inotropic effect of β-adrenergic stimulation. Here, we adopt a 1 Hz stimulation protocol used by Roof et
al. ([40]; Figure 1) to study the effect of 1 μM isoproterenol on isometric contraction and compare it with the effect of increasing
extracellular Ca2 + concentration (Ca2 + o) from 1 mM (control) to 3 mM. As shown in Figure 4A, administration of 1 μM isoproterenol or an increase in Ca2 + oto 3 mM causes a substantial (3-fold) enhancement in peak myoplasmic Ca2 + myo as observed experimentally ([40]; Figure 1). The isoproterenol-dependent effect is a result of PKA mediated dose-dependent increase
(23%) in peak ICa,L current together with an increase (17%) in the maximal uptake rate of the SERCA pump
when compared to the control case (Appendix, Eqns. 57). In the isoproterenol, the
significant increase in peak Ca2 + myo causes an increase in the strength of isometric tension developed (Figure 4B) despite a decrease in myofilament Ca2 + sensitivity. Increase in Ca2 + oto 3 mM causes a similar effect of increasing isometric tension due to an increase
in activator Ca2 + in the cytosol. The increase in peak Ca2 + myo is a result of an increase in SR Ca2 + content due to enhanced Ca2 + entry via ICa,L assisted by impaired Na + /Ca2 + exchange due to elevated Ca2 + o. However, compared to β-adrenergic stimulation, the lack of a decrease in MCS results in a moderately larger
developed force. Figure 4C shows the phase plots of self normalized force versus the instantaneous Ca2 + concentration in the cytosol. As expected, either the presence of isoproterenol or
an increase in Ca2 + ocauses rightward extension (due to increase in peak Ca2 + myo) of the phase plot and increases the area enclosed by it due to elevated myoplasmic
Ca2 + level combined with an increase in tension developed. In the presence of isoproterenol,
the increase in isometric tension developed occurs despite a decrease in myofilament
Ca2 + sensitivity (Figure 4C) which manifests as an increase (0.65 (Figure 4C) which manifests as an increase (0.65 μM to 0.85 μM) in EC50, the average Ca2 + myo at 50% of maximal developed force.

Discussion

Myofilament dynamics have been captured by various representations ranging from the
highly simplified models to complex empirical [41,42] and biophysical models [13]. While simplified models tend to use an explicit parabolic tension profile [43], the empirical models use predefined expressions to specify the average force developed
by the cross bridges, based on experimental observations of isolated muscle contraction
under different loading conditions. On the contrary, biophysical models of cardiac
myofilament dynamics include descriptions of cross-bridge cycling and their elastic
properties. An extensive review of various myofilament models in the literature is
given by Trayanova and Rice [44]. We have developed a composite multiphysics model of excitation-contraction coupling
in the rat ventricular myocyte based on a mechanistic electrochemical model of calcium-induced
calcium-release (CICR) [4] and a detailed mechanochemical model of cooperative activation and crossbridge cycling
[13]. After integrating these component models, the resultant multiphysics model of cardiac
electromechanics is used to examine the mechanisms regulating myofilament contractility
in an isolated rat ventricular myocyte under a voltage clamp protocol.

In particular, we have studied the role of different modulatory factors in influencing
the various types of twitch response (isosarcometric, unloaded and isometric) elicited
by an isolated rat ventricular myocyte. The dependence of isosarcometric contraction
on the amount of activator Ca2 + available in the myoplasm to trigger contraction and the sarcomere length which modulates
myofilament Ca2 + sensitivity is demonstrated (Figure 3). Unloaded cell contraction is investigated to understand the influence of Ca2 + myo transient on the degree of sarcomere contraction achieved by an unrestrained cell
highlighting the enhanced shortening velocity and rate of recovery with increasing
peak Ca2 + myo (shorter time to peak and faster rate of relaxation in the inset in Figure 11B). In agreement with Rice et al. [13] we demonstrate that in isometric contraction (Figure 6), there can be significant internal shortening of the sarcomere as the result of
compliant end connections (low KSE values).

Ca2 + released as a result of CICR is known to act as an actuator, triggering myofilament
contraction by binding to the low affinity regulatory sites on troponin C which act
as the sensor, the Ca2 + affinity of which is a function of dynamically changing sarcomere length as well as
the fraction of strongly bound crossbridges. This hence facilitates not only a feedforward
but also a moderate feedback interaction between the [Ca2 + ]myo transient and the myofilament contractile mechanism. This feedback effect is seen
in Figure 8C where the presence of isometric contraction results in a small decrease in peak
Ca2 + transient, a result of the Ca2 + buffering action of troponin.

Although it is well known that myoplasmic Ca2 + concentration has a direct influence on beat to beat cell shortening, other indirect
modulatory influences exist. Of particular interest is the indirect coupling of the
cellular plasma membrane with the myofilament contractile mechanism. A decrease in
Na + /Ca2 + exchanger function in extruding Ca2 + from the cytosol results in an increased relative role for the SERCA pump, augmenting
SR Ca2 + content. This in turn results in enhanced release increasing the availability of post-release
activator Ca2 + in the cytosol, thus translating into greater degree of contraction (Figure 9). This completes a control loop that allows modulation of NCX activity to force a
readjustment in myofilament contractility.

Experimental conditions such as temperature strongly influence cardiac myofilament
contractility. An increase in temperature results in an increase in sensitivity of
the myofilaments to myoplasmic Ca2 + [22]. As the temperature is increased from 22°C to 30°C, increasing myofilament Ca2 + sensitivity (Figure 10C) causes a moderate increase in peak force despite a decline in peak Ca2 + myo. However, a further increase to body temperature results in a steep decline in force
developed (Figure 10D). Hence, an increase in temperature from 22°C to 38°C which results in a decline
in peak Ca2 + myo[14], causes an overall decrease in force developed. This translates into an overall decrease
in the degree of cell shortening with increasing temperature (Figure 10F). This temperature dependent behavior is not captured by the model proposed by Rice
et al. [13] where an increase in temperature causes an increase in peak force developed as a
result of the large Q10 values used (Qfapp, Qhf, Qhb and Qgxb, Table 1, [13]).

Modulation of myofilament Ca2 + sensitivity (MCS) as a result of a change in Troponin I (TnI) phosphorylation by PKA
has been implicated in heart failure [45]. Here we study the role of MCS in modulating myofilament contractile response (Figure
11) an aspect overlooked in recent modeling studies including Rice et al. [13]. In particular, we model the effect of isoproterenol, a β-adrenergic agonist, which is known to cause a decline in myofilament Ca2 + sensitivity as a result of protein kinase A (PKA) mediated phosphorylation of troponin
I at Ser23/Ser24 [15,16]. Such a decline in Ca2 + sensitivity aids myofilament relaxation in the presence of increased levels of activator
Ca2 + . Figure 4 shows that an increase in amplitude of the myoplasmic Ca2 + transient (a cumulative effect of enhancement in trigger current and increase in uptake
rate of the SERCA pump) is more than adequate to compensate for the decrease in Ca2 + sensitivity, thus facilitating the desired effect of β-adrenergic stimulation, namely an increase in contractile force generated.

Model limitations

1. Our model of a rat ventricular myocyte is limited to Ca2 + related channels, exchanger and pumps (ICa,L, INaCa, IPMCA, Iryrand SERCA pump), while lacking exclusive Na + or K + related channels and transporters (described in Krishna et al. [4]). In our voltage clamp protocol, external solution in the bath is modeled as normal
Tyrode with Cs + substituted for K + . We have represented INaCsin our model (described in Krishna et al. [4]) by the expression for Na + /K + pump formulated by Lindblad et al. [46] replacing K + ion concentrations with the Cs + ion concentration. While ensuring whole cell Na + ion balance, the peak Na + /Cs + pump current is modified to be one-sixth to account for the decreased potency of the
cation Cs + in activating the pump. The voltage-dependence of INaCsis adopted from the data on Na + /K + pump from Hansen et al. [47]. The role of mitochondrial Ca2 + uniporter is not modeled in this study due to its negligible (< 1%) role in Ca2 + transport from the cytosol [48]. This model is aimed at mimicking voltage clamp conditions where channels other than
calcium are blocked, and it cannot be used to study any action potential-induced Ca2 + transient-frequency relationship. However, its focus on the Ca2 + dynamics allows one to comprehend more clearly the important role of Ca2 + signalling pathways and feedback control systems in maintaining whole cell homeostasis
over a prolonged period of time.

2. The cooperative activation of the thin filament and the strain-dependent transitions
of the crossbridge cycle have been approximately modeled as non-spatial, state-variables.
However, this simplification is valid as these transitions are inherently local phenomena
and the model reproduces a wide range of steady state and dynamic responses in cardiac
muscle [13]. Although a CaM-dependent pathway is reported [18] to be responsible in modulation of myofibrillar contractility implying a possible
CaM mediated role for Ca-dependent kinases or phosphatases in regulating myofilament
contractility (particularly in frequency dependent acceleration of relaxation), further
studies are required to clarify the molecular mechanisms involved.

3. The temperature dependence of passive force attributed to titin and other cytoskeletal
elements is not modeled in this study and the assumption of constant stiffness for
the series elastic element (KSE) does not account for temperature dependent effects.
This is an area where the model can be expanded but additional measured data on the
temperature sensitivity of these elements is necessary. Regardless, our model provides
reasonable approximations to the temperature dependence of developed force in intact
thin rat ventricular trabeculae [14].

Conclusion

We have developed a composite mathematical model for cardiac electromechanics which
includes a detailed description of Ca2 + dynamics under voltage clamp conditions in the rat ventricular myocyte, based on experimental
data [4,13]. We have investigated the role of different factors including the myoplasmic Ca2 + myo transient and the sarcomeric length in influencing various types of twitch responses
obtained under different loading conditions (including isosarcometric, isometric and
unloaded conditions). Various control loops influencing cell shortening have been
explored. In particular, the bidirectional interaction of the Ca2 + transient with the myofilament contractile mechanism and the importance of indirect
SR mediated interaction of the sarcolemma with the contractile machinery is highlighted
by showing the regulation of isometric contraction by the degree of NCX activity.
The effect of temperature on cell shortening is investigated identifying the differential
sensitivity of the Ca2 + myo transient and the myofilament contractile mechanism. The important role of myofilament
Ca2 + sensitivity in force generation is studied with particular emphasis on the effect
of β-adrenergic stimulation on cardiac contractile response. In agreement with Janssen
[37], we also demonstrate a key linear relationship between the rate of contraction and
relaxation, which is shown here to be intrinsically coupled over the full range of
physiological perturbations (including temperature, sarcomeric length, activator Ca2 + , and isoproterenol; e.g. see insets in Figures 10D, 3A-ii).

This study demonstrates that the model has long-term stability in regulating myoplasmic
Ca2 + , as shown in the 18-sec duration experiments at a physiological rate of stimulation
(conditioning train of voltage clamp pulses at 5 Hz) shown in Figure 9. This long term Ca2 + balance under physiological conditions is crucial in facilitating implementation of
this model in large scale simulations such as frequency dependent studies analyzing
cellular force-frequency response. Our study also provides mechanistic insights into
whole-cell responses to a wide variety of testing approaches used in studies of cardiac
myofilament contractility that have appeared in the literature over the past two decades
(Figures 3,4,5,6,7,8,9,10 and 11). Thus, the model serves as a platform for the predictive modeling of VC investigations
of cardiac electromechanics pertaining to the rat ventricular myocyte in a number
of areas. These are fundamental issues that would benefit from a better mechanistic
understanding of the cardiac contractile mechanism in the rat ventricluar myocyte.
This study is aimed at providing an initial step towards this goal.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AK developed the coupled electromechanical model of the rat ventricular myocyte, carried
out the voltage clamp modeling studies, and drafted the manuscript. MV made substantial
intellectual contributions to the study and in drafting of the manuscript. PTP made
intellectual contributions to the manuscript as well as significant contributions
to the drafting of the manuscript. JWC made key contributions to the conception of
the study, design, analysis and interpretation of results, and drafting of the manuscript.
All authors read and approved the final manuscript.

Acknowledgements

This work was supported by a research grant from Methodist Hospital Research Institute.
The authors would like to thank Liang Sun for his early contributions to this work.

References

Fabiato A, Fabiato F: Contractions induced by a calcium-triggered release of calcium from the sarcoplasmic
reticulum of single skinned cardiac cells.